Any Python code examples of bmesh operations on multiple objects?
This is an example based upon batFINGER's link that uses a vertex index. 

import bpy
import bmesh

context = bpy.context
mesh_obs = [o for o in context.selected_objects if o.type == 'MESH']
bm = bmesh.new()

for ob in mesh_obs:
    me = ob.data
    bm.from_mesh(me)

    bm.verts.ensure_lookup_table()
    bm.verts.remove(bm.verts[0])

    bm.to_mesh(me)
    me.update()
    bm.clear()